question_answer

If\[(a,\text{ }{{a}^{2}})\]falls inside the angle made by the lines \[y=\frac{x}{2},\text{ }x>0\]and\[y=3x,\text{ }x>0,\]then a belongs to       AIEEE  Solved  Paper-2006

A) \[(3,\infty )\]    

B)        \[\left( \frac{1}{2},3 \right)\]     

C)        \[\left( -3,-\frac{1}{2} \right)\]  

D)        \[\left( 0,\frac{1}{2} \right)\]

Correct Answer: B

Solution :

The graph of equations\[x-2y=0\]and\[3x-y=0\]is as shown in the figure. Since, given point \[(a,\,\,{{a}^{2}})\] lies in the shaded region. Since,        \[x>0\Rightarrow a>0\]               ...(i) From given condition, \[{{a}^{2}}-\frac{a}{2}>0\] \[\Rightarrow \] \[a\left( a-\frac{1}{2} \right)>0\] \[\Rightarrow \]\[a<0\]or \[a>\frac{1}{2}\]                                   ...(ii) Also, \[{{a}^{2}}-3a<0\] \[\Rightarrow \]\[a(a-3)<0\] \[\Rightarrow \]\[a\in (0,3)\]                                                  ...(iii) From Eqs. (i), (ii) and (iii). we get \[a\in \left( \frac{1}{2},3 \right)\]